For example, without the One Ring we wouldn't have Nazgūls, nor their influence in their respective kingdoms.

Let's assume the beginning and the middle would have gone as you described. No matter prepared Sauron would have had time to become, he still would have been bested by the Nśmenorians etc.

I believe Sauron would have been able to flee the Sinking even without the Ring. It's the opposite: Tolkien questioned himself how Sauron could save the Ring from the Sinking. And answered that he must have carried it with himself as he flee in his bodyless spirit form. After all, he was a Maia and therefore capable of controlling physical matter with his mere Fėa.

The War of the Last Alliance... Sauron would have been defeated, again. But would he have had enough power left to rebuild himself, no matter how slowly, without the Ring existing somewhere? I'm afraid my Ringlore is quite rusty, and someone else will have to answer this one.

What I know is that IF Sauron would have survived these destroyings of his body (yes, would he have surived its destroyment in the Fall of Nśmenor without the Ring?) and made it so far as to began the War of the Ring... which obvioysly wouldn't have been called that... he would have won.

And now someone who remembers their LotR-fu better than me can come and correct this mess.

Without the One Ring, the Nine would have still been powerful artifacts and their use might have still resulted in their owners being transformed into Nazgūl, but Nazgūl not subject to Sauron's will. The Seven Rings of the Dwarves would have still been just as troublesome, unable to subjugate the Dwarf-lords, but still having an effect upon them. Of course, the Elven Rings could have been used with impunity with Sauron unable to affect them. Actually, without the Master Ring, Sauron might have been undefeatable--except perhaps by the power of the Three Rings.'There are older and fouler things than Orcs in the deep places of the world.' - Gandalf the Grey, The Fellowship of the Ring